I originally posted this on the Mathematics Stack Exchange, thinking that the best place to post it, but the question quickly accumulated a bunch of close votes since it was not quite within the scope of mathematics they want there. (I guess either because it was primarily subjective, or perhaps focused on a singular instance, or because it was more focused on the educational aspect of things.) If this isn't the right sort of place, or this is just a bad question, let me know and I'll stop bugging people - this has just been a source of anxiety for a long time now.

This is the next-best place I could think to post this, in which, in short, I ask about how the grades of a special topics course in category theory at my university reflect on us the students, the lecturer, and the material, and more importantly would could have been done better.

Foreword: I want to preface this that this is based on an actual example of what happened in a course of mine this past semester, wherein I'm speaking from the perspective of a student. It's probably highly subjective and there's probably no clear-cut correct answer, because it's a complicated matter in its own right. Me asking this question was mostly motivated by how this has been a question I've mulled over for months, and even more intensely ever since our final exam grades were announced, and partially spurred on by a MSE question on curved/scaled grades.

In any event, this will be a bit of a long remembrance on the course. While it's a bit lengthy, I think each bit has its own importance, as I touch on when I formally state my question and hypotheses at the end, so obligatory wall of text warning.

Context: This happened during the course of the past fall semester, fall of $2018$. At our university, they allow teachers/professors to teach "special topics" courses, typically aimed at senior undergraduate students majoring in mathematics (or so I can gather by the course number being $490$). They're all collected under the $490$ number and they're generally not taught frequently - we're talking a semester or so "special event" sort of deal, as opposed to a more regular course like real analysis or number theory that is offered once/twice a year.

One of our lecturers, who primarily lectures in set theory and algebraic structures, is very passionate about category theory, and decided to teach a course on it despite having never done so before. (It might even be the first time in our university's history.) With a sufficient number of people joining the course (slightly over a dozen), the course began that August.

My Experiences With Him Thus Far: I want to emphasize that this lecturer is a damn fine lecturer. I've had him for a set theory course and this category theory course, and I'll be taking a more advanced "special topics" course in algebraic structures with him next semester as well. Most of the students seem to enjoy him, and it's not because he's an "easy A" type of professor - really, he's anything but, his courses are hard, bordering on hellish at times. He covers a lot of material, and goes very fast, but that's part of why I like him so much, because he challenges you like that. We have one professor who is more of the "easy A" type who teaches algebraic structures, for example; she mostly spends whole class sections working examples from the textbook, whereas what we covered in one semester with her, basically boiled down to a week or two with this lecturer. He also really cares about student understanding, enough to completely upheave his own plans for the rest of the course of the semester if it seems like we're not understanding something, just to make sure we understand it damn well. Plus he just has an overall wonderful, kinda nerdy personality and is clearly passionate about the material, which makes for an enjoyable experience all around even ignoring the math stuff.

So to say he's my favorite professor is a bit of an understatement, considering his set theory class really helped me get out of a depressive spell and educational rut, and also helped to cement my decision to pursue my major in math.

With that out of the way...

The Category Theory Course From Hell: ... the course did not go particularly well for nearly anyone. In a broad overarching look at the course, I don't feel like we covered nearly as much as was desired. Our lecturer mostly covered the following topics:

  • Some basic groundwork: what's a category, what's a morphism, what's a functor, some general motivations for the study of category theory
  • Constructs and morphisms: we studied various special morphisms (epis, monos, etc.), and also special constructs (namely products, equalizers, pullbacks, and some others)
  • Duality, and the consequences of it
  • More on functors, particularly considering limits, diagrams, and cones
  • A very basic overview of natural transformations (we were pressed for time)
  • Yoneda's lemma

"Is This Enough?" One of my lurking worries is that this might be too little material, but looking around this might be sufficient for a first course? We've covered almost everything aside from adjoints as recommended in this question on MSE about topics for a first course in category theory. So I suppose our lecturer was pretty spot-on with his goals?

The material was mostly presented in the order above, interlaced with a fair number of examples from the text and relations with various algebraic structures.

How It Could Have Gone: I want to take this moment to digress on what my professor had alternately in mind for the course, a sort of alternative outline for the material. I don't have it formally or in any detail, but he felt that the material would come off easier for us in the above order, instead of describing everything in the language of categories, functors, limits, natural transformations, etc., right off. I think most of the class agreed that it was easier for us in this respect since I feel like the ball just dropped with a lot of us when we hit natural transformations.

I mostly am just stating this in case anyone feels it a relevant factor in what occurred.

I believe the lecturer had other topics in mind he wanted to cover for the course; adjoints came up often in particular when discussing "what we're going to eventually cover." I'm not sure if anything else beyond that was intended.

We mostly were getting backed up on not understanding the material too well. I believe we spent a good extra week laying down pat what the universal properties of the various constructs represented, for example. This became evidenced in our exam grades: there were two midterms and a final in this course. Homework was also a factor in this course as well.

Typically this lecturer's courses follow the standard grading scale:

  • A: $90\%$ or higher
  • B: $80\%$ to $90\%$
  • C: $70\%$ to $80\%$
  • D: $60\%$ to $70\%$
  • F: Less than $60\%$

I think it was after the first midterm or the second that the grading scale got shifted. He said this was in part because the course was experimental, i.e. his goal was not to drag us in here and fail us, and averages were miserable, somewhere in the $50$'s. The shifted scale operated in intervals of $20\%$:

  • A: $80\%$ or higher
  • B: $60\%$ to $80\%$
  • C: $40\%$ to $60\%$
  • D: $20\%$ to $40\%$
  • F: Less than $20\%$

On top of this, each midterm was heavily curved with some extra bonus points.

I forget how bad the results were for the midterms since these were discussed in-class - detailing how many got which grade and what the average was and all that, for an overall idea of how we performed. So I don't remember the details too well. I just know that the midterms got a healthy boost in addition to the shifted overall grading scale: the post-curve average on the midterms was each about $71\%$ for reference, so I imagine (remembering my own uncurved grades) the averages were in the $60$'s or $50$'s.

And again, I want to preface that this was completely out of left-field from my experience with this lecturer. This not at all how things normally go for him, and I think that point was emphasized by him and other students (who had him in other set theory classes than mine, or had him for algebraic structures while I had that easy-A professor). I never would've imagined him basically changing a passing grade to a $40\%$ at any point up to now.

The final exam was the real kicker for me in particular - I have ranted about this to no end to my friends on Discord. The lecturer decided to take it easy on us and the exam, rather than be mostly proofs, was $70\%$ definitions of things, i.e. you could pass just by knowing what the definitions were. (The midterms were not like that, they were exceptionally proof-heavy.) The remainder was a few proofs, I forget the details on that $30\%$. We got an email this time detailing our performance on the final exam.

And keep in mind, this is under the adjusted scale:

  • $3$ made A's (in full disclosure, I was among them)
  • $2$ made B's
  • $1$ made a C
  • $0$ made D's
  • $6$ made F's
  • From what I can determine based on the class roster, two people did not attend the final (and by definition failed it and the course outright per university policy - probably the same two students who were perpetually absent from the class lectures)
  • The average was $59\%$, again after some curving (and also the test also had a few smaller bonus problems on it, so the average was actually possibly way worse)

Anecdote: To memory, this also reflects the general distribution of grades for the midterms as well, in that a lot of people either did well or failed, with little in the middle. That is to say, it is like an inverted bell curve, where most of the data in the distribution is at either extreme, as opposed to around the mean. I don't recall the exact numbers though, this is just a rough idea from my memory.

Why Call It The Course From Hell?: After all, I got an A, why am I complaining?

Well, for one, we look at the grade distribution. While I don't know everyone's final grade, it's already obvious two people failed the class, and probably half the class or more may have as well considering that they failed a test mostly on definitions. That's unusual for this lecturer - no, it's unusual for everything I've seen this far in my academic career. I've never seen half the class or more fail a course and that's despite taking nearly every upper-level math course at my university (plus some in other subjects). It is a huge outlier in that respect.

On the other, it was difficult as hell, even coming from someone who made that A. I have lost so much sleep this past semester from this course and was pushed to a breaking point several times when it combined in a perfect storm with my other coursework. My A was around a $90\%$ or so - just right on the edge of A under a proper grading scale, although this is with curves in mind, my "uncurved" grade is probably in the mid-to-low $80$'s. And that much I was lucky to get - I had to work tooth and nail, devoting probably $3-4$ times the amount and time of work for this course as I did my others combined. My category theory questions on MSE probably reflect some of that because, looking back, I feel a little stupid for asking some of them, and others make me think about how basic the material is, considering we didn't seem to go as deep into the material as desired.

And of course, I don't think anyone expected that we'd be held back by our own lack of understanding and thus not cover as much as the lecturer wanted. So that's a third reason.

So, Finally, The Big Question:

How does all of this reflect on the lecturer, and on us as students?

I feel like this reflects poorly on all of us, or perhaps it instead is a reflection of the material? My own thoughts on the matter:

  • Part of me wants to say that the lecturer wasn't at wrong here. I feel like he did a decent job - as good as he could have anyhow. But at the same time, I was one of the higher grades in the course, and I think that takes away the right for me to judge him fairly since some people simply "got it" (someone was getting perfect scores on everything), yet most did not (ergo half the class failing). That I was fairly above-average means that I am somewhat in the realm of "I just ... get it", that even though I had a hellish time with the material, some people struggled way, way more than I did. So I don't have the fairest shake in that respect.

  • Part of me wants to say that it lies in part on the material. After all this, the main thing I learned was that ... category theory is hard. Abstracting mathematics like this makes it hard to penetrate and understand, and I understand that I only have basic knowledge of the material. I did not develop an intuitive grasp on the material like I have other subjects. Part of me wants to blame that on the material itself, in that it just might not be compatible with me, that I might just not be able to think in a categorical sense, or at least not particularly well. (It's sort of like how everyone has their specialties - maybe category theory is just not at all one of mine, arguably the same for everyone in the course. And therefore, being incompatible with the material, we do poorly than some people who are more compatible.) This begs a question all its own - for those who took courses in category theory in undergrad, or related courses (e.g. algebraic topology), how did people fare in your case? Perhaps this is normal and I'm just overthinking it.

  • Part of me thinks I don't want to blame the lecturer, and recognizing that bias (because, damn, I love him as a lecturer) I look at the professor himself. Part of me thinks that people only did as well as they did because of him - both good and bad. Perhaps he should have actually gone slower through the material? (Again, this begs the question of how other courses handled it.) Perhaps the altered grading scale actually encouraged people to grow lazy and not do as much? Or perhaps was his approach altogether wrong - that maybe he should've simply done it all in terms of functors and natural transformations as he intended?

  • And of course, part of me blames the students. I don't think it's a lack of expertise on our part - we were all seniors (maybe one or two juniors?) and majoring in mathematics. But something is wrong when you fail an exam that requires you to basically recite definitions to pass. Perhaps the lecturer wasn't clear enough with some definitions (which touches on various previous points), or perhaps some students weren't putting in the necessary effort (again touching on a number of previous points). I mean, that final exam - both in its formulation, and in terms of everyone's performance - alone is the symptom of some larger problem, wouldn't you say?

And My Questions To You:

  • In your opinion, what was the cause of this, and how could it have been avoided, if at all? I've offered my opinions and experiences above, but I'm a highly biased individual in too many ways to count in this case. I'd really appreciate some secondary viewpoints because I know this lecturer is particularly passionate about the material, so if I end up a TA in grad school I would like to help him with this. And of course, there's also my own curiosity because this is new to me.

  • How have category theory courses gone in your experience - or if not "purely" category theory, things directly tied to it like algebraic topology?

  • In your opinion, how does this reflect on the students, the lecturer, and the material? Like I said, I'm biased - maybe we all just suck, maybe the lecturer sucks, you know what I'm saying? Or maybe this is a natural phenomenon, like this is just how hard category theory courses tend to be. Obviously I doubt it's so clear-cut but this has been bugging me for the longest time.

  • Should the grades have been curved? While it's a question slightly tangential to all of the other questions, I do wonder if this is a fair practice, in that our grades were curved and rescaled and all a lot, mostly just because the lecturer didn't want to fail us and hurt our GPAs for signing up for an experimental course. And I don't think any such sort of interference has occurred in our grades thus far, or at least there wasn't this much curving and such in my past courses. Which begs the question: did our lecturer go too far in that respect? (This hints at another worry in that some of us who passed, may not have deserved it. I don't totally feel like I did because of my lack of an intuitive understanding of the material, even though I put in hard work.)

  • I guess if there's anything else you feel of value to say, that isn't strictly related to my main questions, I'd like to hear it as well.

And thanks for taking the time to read this insane wall of text, if you've made it this far. Even just listening to my ramble is well-appreciated. :)

Of course, if there's any questions you feel like I need to answer myself - further context you think might help - I can try to help where necessary.

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    $\begingroup$ Honestly, in similar special topic type courses I've seen it is not uncommon for everyone who participates even marginally to pass. The fact that half of your class earned an F, well, this guy is pretty serious. I really think you should be happy what you learned and not worry about grades of other students. There is really no good end to this line of thought. Furthermore, grades are merely tools to get you to think. You're thinking. Hard work is not for the grade or beating other kids, it's for the edification of your own intellect etc. This lesson I wish I had learned much earlier... $\endgroup$ – James S. Cook Dec 26 '18 at 7:50
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    $\begingroup$ The examples you described (preorders and ordinals) frankly sound boring and are not good motivating reasons to study category theory. It seems like you did not actually do anything with category theory. Did the instructor ever solve a worthwhile problem not about category theory using the machinery of category theory? I don't think the audience had the background to appreciate the point of this stuff, like trying to teach set theory to 3rd graders. I agree with Cook that you should not worry about how other people did. $\endgroup$ – KCd Dec 26 '18 at 13:28
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    $\begingroup$ The question is absurdly long, and I can't find anything in there that is actually a question. $\endgroup$ – Ben Crowell Dec 28 '18 at 2:12
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    $\begingroup$ I think there are several good questions hidden in this one - for example, is it reasonable for an instructor to change grading policies during the realization of a course? or, what level of improvisation regarding course contents is permissible? But these need to be separated out from an overly long text and too many purely personal details that possibly obfuscate the essential points. $\endgroup$ – Dan Fox Dec 28 '18 at 10:01
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    $\begingroup$ I could certainly post the questions concerning grading/curving changing mid-course as separate questions if desired. I just don't think I could offer much of a viewpoint other than what I've seen as a student in those respects; SE would get far more out of someone with experience doing so. Similar for improvisation - not being an educator kinda skew my viewpoint. $\endgroup$ – Eevee Trainer Dec 28 '18 at 10:04

This sounds like the same mistake I made many years ago, namely trying to teach category theory to a class in which many students had not seen many of the specific examples that make category theory interesting and memorable.

For example, left adjoints of forgetful functors are a wonderful idea, unifying notions of free algebraic structures, abelianizations of groups, universal enveloping algebras of Lie algebras, Stone-Cech compactifications of Hausdorff spaces, and other things. But that's of little value if students haven't seen those notions before.

I was, fortunately, teaching a graduate course, where most students had seen at lest a little bit of the background material. But that little bit was nowhere near enough for a really successful course. From your description, it appears that your teacher got himself (and you) into an even worse situation, in that his students had even less of the desired background than mine did.

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    $\begingroup$ "For example, left adjoints of forgetful functors are a wonderful idea, unifying notions of free algebraic structures, abelianizations of groups, universal enveloping algebras of Lie algebras, Stone-Cech compactifications of Hausdorff spaces, and other things. But that's of little value if students haven't seen those notions before." -- Considering I've seen none of those ... yikes... (Granted we didn't go over adjoints though he intended to originally.) $\endgroup$ – Eevee Trainer Dec 27 '18 at 3:14
  • $\begingroup$ At least he plans to go briefly over free algebraic structures in a course next semester (a sort of deeper dive than our courses normally go into for algebraic structures). But at this point - between your answer and some of the discussion in the comments of the question - I'm definitely seeing why some people in the class might've gone "why are we doing this in the first place?" $\endgroup$ – Eevee Trainer Dec 27 '18 at 3:14

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